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The rectifiability of entropy measures in one space dimension


De Lellis, C; Rivière, T (2003). The rectifiability of entropy measures in one space dimension. Journal de Mathematiques Pures et Appliquees, 82(10):1343-1367.

Abstract

We show that entropy solutions to 1-dimensional scalar conservation laws for totally nonlinear fluxes and for arbitrary measurable bounded data have a structure similar to the one of BV maps without being always BV. The singular set—shock waves—of such solutions is contained in a countable union of C1 curves and ℋ 1 almost everywhere along these curves the solution has left and right approximate limits. The entropy production is concentrated on the shock waves and can be explicitly computed in terms of the approximate limits. The solution is approximately continuous ℋ 1 almost everywhere outside this union of curves.

Abstract

We show that entropy solutions to 1-dimensional scalar conservation laws for totally nonlinear fluxes and for arbitrary measurable bounded data have a structure similar to the one of BV maps without being always BV. The singular set—shock waves—of such solutions is contained in a countable union of C1 curves and ℋ 1 almost everywhere along these curves the solution has left and right approximate limits. The entropy production is concentrated on the shock waves and can be explicitly computed in terms of the approximate limits. The solution is approximately continuous ℋ 1 almost everywhere outside this union of curves.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:conservation laws; entropy solutions; measurable bounded data; concentration
Language:English
Date:2003
Deposited On:29 Nov 2010 16:26
Last Modified:06 Dec 2017 20:49
Publisher:Elsevier
ISSN:0021-7824
Publisher DOI:https://doi.org/10.1016/S0021-7824(03)00061-8
Related URLs:http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1037.35044

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